Question

6. Given the following function TR(R) 6000 -40 TCQ) 1000 +10,200 a. Find the revenue maximizing level of output (check 2d order conditions also), b. Find the maximum revenue c. Find the corresponding profit function d. Find the profit maximizing level of output (check 2nd order conditions also) e. Find the maximum profit 7. Given the following function TR(x)-3x260x TC(x) = 5,500 + 50x a. Find the revenue maximizing level of output check 2d order conditions also), b. Find the maximum revenue c. Find the corresponding profit function d. Find the profit maximizing level of output (check 2nd order conditions also) e. Find the maximum profit 8. Indicate whether the following functions are (1) increasing or decreasing and (2) concave or co nvex at the given points f(x) 5x2 + 6x-89, at x-3 f(x) = 7x3 + 7x2-15x, atx=-4 (a) (b) (c) f(x) =-7x2 + 14x-24, at x= 1 9. Optimize the following functions and test the second-order conditions at the critical points to distinguish between a relative maximum and a relative minimum. a. b. y у 8x2-208x + 73 3x3-45x2-675x + 13 10. Find the marginal and average functions for each of the following total functions: TR(Q)-202960 b. TC() 503067 11. Given that total cost can be expressed as TC(x) x3-140x2 4200, find: a the average cost fimction b. the level of output that minimizes AC (check 2d order conditions also) c. the value of average cost at its minimum level

Answer 1

Question 6.
a. $\frac{\mathrm{d} TR}{\mathrm{d} Q} = 600 - 8Q = 0$
So, 8Q = 600
So, Q = 600/8 = 75
So, Q = 75

Second order condition: $\frac{d^{2}TR}{dQ^{2}} = -8 < 0$
So, TR is maximum at Q = 75.

b. Maximum revenue = $600Q - 4Q^{2} = 600(75) - 4(75^{2}) = 45,000 - 22,500 = 22,500$

c. Profit = TR - TC = $600Q - 4Q^{2} - (100Q&plus;10,200) = 600Q - 4Q^{2} - 100Q - 10,200$
So, $\pi = - 4Q^{2} &plus; 500Q - 10,200$

d. $\frac{\mathrm{d} \pi }{\mathrm{d} Q} = -8Q &plus; 500 = 0$
So, 8Q = 500
So, Q = 500/8 = 62.5

Second order condition: $\frac{d^{2}\pi }{dQ^{2}} = -8 < 0$
So, profit is maximized at Q = 62.5

e. Maximum profit =  $\pi = - 4Q^{2} &plus; 500Q - 10,200$ = $-4(62.5)^{2} &plus; 500(62.5) - 10,200 = -15,625 &plus; 31,250 - 10,200 = 5,425$

Maximum profit is 5,425.

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